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Properties

Label 2-42e2-12.11-c1-0-3
Degree $2$
Conductor $1764$
Sign $-0.624 - 0.780i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.13 + 0.850i)2-s + (0.554 − 1.92i)4-s − 3.87i·5-s + (1.00 + 2.64i)8-s + (3.29 + 4.37i)10-s − 3.46·11-s − 0.296·13-s + (−3.38 − 2.12i)16-s + 1.56i·17-s + 7.07i·19-s + (−7.43 − 2.14i)20-s + (3.92 − 2.95i)22-s − 5.43·23-s − 9.98·25-s + (0.335 − 0.252i)26-s + ⋯
L(s)  = 1  + (−0.799 + 0.601i)2-s + (0.277 − 0.960i)4-s − 1.73i·5-s + (0.356 + 0.934i)8-s + (1.04 + 1.38i)10-s − 1.04·11-s − 0.0822·13-s + (−0.846 − 0.532i)16-s + 0.380i·17-s + 1.62i·19-s + (−1.66 − 0.479i)20-s + (0.835 − 0.628i)22-s − 1.13·23-s − 1.99·25-s + (0.0657 − 0.0494i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 - 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.624 - 0.780i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.624 - 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2436525863\)
\(L(\frac12)\) \(\approx\) \(0.2436525863\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13 - 0.850i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 3.87iT - 5T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 + 0.296T + 13T^{2} \)
17 \( 1 - 1.56iT - 17T^{2} \)
19 \( 1 - 7.07iT - 19T^{2} \)
23 \( 1 + 5.43T + 23T^{2} \)
29 \( 1 + 6.85iT - 29T^{2} \)
31 \( 1 - 2.81iT - 31T^{2} \)
37 \( 1 - 2.51T + 37T^{2} \)
41 \( 1 - 3.55iT - 41T^{2} \)
43 \( 1 - 0.682iT - 43T^{2} \)
47 \( 1 + 2.36T + 47T^{2} \)
53 \( 1 + 0.623iT - 53T^{2} \)
59 \( 1 + 8.85T + 59T^{2} \)
61 \( 1 - 2.66T + 61T^{2} \)
67 \( 1 - 10.6iT - 67T^{2} \)
71 \( 1 + 0.539T + 71T^{2} \)
73 \( 1 - 7.39T + 73T^{2} \)
79 \( 1 - 6.16iT - 79T^{2} \)
83 \( 1 - 6.15T + 83T^{2} \)
89 \( 1 - 11.7iT - 89T^{2} \)
97 \( 1 + 6.84T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.631022810847228656936152655099, −8.520204791014844823736380522506, −8.123745496188986542044772383925, −7.65155513255029154083886718847, −6.22542588605244892099956984980, −5.64630101617410768147387951600, −4.90871858564224764603632702404, −4.00883700604039858471023355979, −2.19776007444181731637267097585, −1.20098408421241316009927911398, 0.12335289185746413892489917120, 2.11747418113496779458162954828, 2.79859130938275010303303445592, 3.49843000118979788198929689216, 4.75194273579955211479556250311, 6.07473839888442792112259701861, 6.95056002884379837874105139068, 7.42856015501733564194264582363, 8.170936835580030793665567676569, 9.220508680365711369566512050661

Graph of the $Z$-function along the critical line